Homogeneous submanifolds and isoparametric hypersurfaces in symmetric spaces of non-compact type

Zusammenfassung

In applied science, it is common that the maximum number of variables that can be handled on problems with certain complexity is not greater than three or four. This is because the amount of data increases as a power of the size, and current computers have a very reduced capacity to store multidimensional data sets. However, the range of problems that can be solved effectively is often greater. Theoretical science has answered numerous practical questions over the years. This is because simplifying assumptions are made and therefore it is necessary to demonstrate mathematically that these simplifications do not have excessive influence on the solution of the problem. On the other hand, another method for solving problems is to take into account the symmetries in order to reduce the number of degrees of freedom. Symmetry in science is ubiquitous, but we need to understand this in a more general way. There are not only symmetries in certain objects and physical forms, but they also appear in equations and theoretical constructions. The Nobel Laureate P.W. Anderson, for example, claimed that "it is only slightly overstating the case to say that physics is the study of symmetry". In fact, in mathematics, the symmetry of an object is the group of transformations of that object that leave it invariant. Thus, the theory of groups is the language of many areas of science. We are therefore entering in the field of continuous groups of transformations, which are known as Lie groups. In this thesis project, we are interested in Riemannian manifolds, which are the mathematical object that models the Theory of General Relativity. The group of symmetries of a Riemannian manifold is called the Group of Isometries, and when a group of isometries acts on a manifold we talk about an isometric action. The orbits of an isometric action are the subspaces that are invariant under the action. We are interested, but not exclusively, in polar actions, first by their intrinsic interest, but also by their practical interest. This allows one to define coordinates adapted to physical or mathematical problems. Specifically, we intend to study the orbits of polar actions in symmetric spaces (although in this case the adjective "symmetric", has a particular meaning and refers to spaces that support an involutive isometry at each point). The geometric criteria that will be used to characterize these orbits will be the isoparametric property, although we must have other possibilities in mind such as the study of hypersurfaces with constant principal curvatures. In fact, the first problem that we have tackled recently has been the study of the isoparametric hypersurfaces in complex hyperbolic spaces. It is known that there are non-homogeneous examples, but they still have a high degree of symmetry that we intend to characterize using the Hopf mapf.